Find largest d in array such that a + b + c = d

Given a set S (all distinct elements) of integers, find the largest d such that a + b + c = d
where a, b, c, and d are distinct elements of S.

Constraints:
1 ≤ number of elements in the set ≤ 1000
INT_MIN ≤ each element in the set ≤ INT_MAX

Examples :

Input : S[] = {2, 3, 5, 7, 12}
Output : 12
Explanation: 12 is the largest d which can be represented as 12 = 2 + 3 + 7

Input : S[] = {2, 16, 64, 256, 1024}
Output : No solution

Method 1(Brute Force)
We can solve this problem using simple brute force approach which is not very efficient as |S| can be as large as 1000. We’ll sort the set of elements and start by finding the largest d by equating it with the sum of all possible combinations of a, b and c.
Below is the implementation of above idea :

C++

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// CPP Program to find the largest d
// such that d = a + b + c
#include <bits/stdc++.h>
using namespace std;
  
int findLargestd(int S[], int n)
{
    bool found = false;
  
    // sort the array in
    // ascending order
    sort(S, S + n);
  
    // iterating from backwards to 
    // find the required largest d
    for (int i = n - 1; i >= 0; i--) 
    {
        for (int j = 0; j < n; j++) 
        {
  
            // since all four a, b, c, 
            // d should be distinct
            if (i == j)
                continue;
  
            for (int k = j + 1; k < n; k++) 
            {
                if (i == k)
                    continue;
  
                for (int l = k + 1; l < n; l++) 
                {
                    if (i == l)
                        continue;
  
                    // if the current combination  
                    // of j, k, l in the set is 
                    // equal to S[i] return this 
                    // value as this would be the 
                    // largest d since we are
                    //  iterating in descending order 
                    if (S[i] == S[j] + S[k] + S[l])
                    {
                        found = true;
                        return S[i];
                    }
                }
            }
        }
    }
    if (found == false)
        return INT_MIN;
}
  
// Driver Code
int main()
{
    // Set of distinct Integers
    int S[] = { 2, 3, 5, 7, 12 };
    int n = sizeof(S) / sizeof(S[0]);
  
    int ans = findLargestd(S, n);
    if (ans == INT_MIN)
        cout << "No Solution" << endl;
    else
        cout << "Largest d such that a + b + "
             << "c = d is " << ans << endl;
    return 0;
}

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Java

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// Java Program to find the largest
// such that d = a + b + c
import java.io.*;
import java.util.Arrays;
  
class GFG 
      
// function to find largest d 
static int findLargestd(int []S, int n)
{
    boolean found = false;
  
    // sort the array in
    // ascending order
    Arrays.sort(S);
  
    // iterating from backwards to
    // find the required largest d
    for (int i = n - 1; i >= 0; i--) 
    {
        for (int j = 0; j < n; j++)
        {
  
            // since all four a, b, c, 
            // d should be distinct
            if (i == j)
                continue;
  
            for (int k = j + 1; k < n; k++) 
            {
                if (i == k)
                    continue;
  
                for (int l = k + 1; l < n; l++) 
                {
                    if (i == l)
                        continue;
  
                    // if the current combination  
                    // of j, k, l in the set is 
                    // equal to S[i] return this 
                    // value as this would be the 
                    // largest d since we are  
                    // iterating in descending order 
                    if (S[i] == S[j] + S[k] + S[l])
                    {
                        found = true;
                        return S[i];
                    }
                }
            }
        }
    }
    if (found == false)
        return Integer.MAX_VALUE;
  
    return -1;
}
  
// Driver Code
public static void main(String []args) 
    // Set of distinct Integers
    int []S = new int[]{ 2, 3, 5, 7, 12 };
    int n = S.length;
  
    int ans = findLargestd(S, n);
    if (ans == Integer.MAX_VALUE)
        System.out.println("No Solution");
    else
        System.out.println("Largest d such that "
                         "a + " + "b + c = d is "
                                             ans );
          
}
  
// This code is contributed by Sam007

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Python3

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# Python Program to find the largest 
# d such that d = a + b + c
  
def findLargestd(S, n) :
    found = False
  
    # sort the array in ascending order
    S.sort()
  
    # iterating from backwards to 
    # find the required largest d
    for i in range(n-1, -1, -1) :
        for j in range(0, n) :
  
            # since all four a, b, c, 
            # d should be distinct
            if (i == j) :
                continue
  
            for k in range(j + 1, n) :
                if (i == k) :
                    continue
  
                for l in range(k+1, n) :
                    if (i == l) :
                        continue
  
                    # if the current combination 
                    # of j, k, l in the set is 
                    # equal to S[i] return this 
                    # value as this would be the 
                    # largest d since we are 
                    # iterating in descending order
                    if (S[i] == S[j] + S[k] + S[l]) :
                        found = True
                        return S[i]                 
  
    if (found == False) :
        return -1
  
# Driver Code
  
# Set of distinct Integers
S = [ 2, 3, 5, 7, 12 ]
n = len(S)
  
ans = findLargestd(S, n)
if (ans == -1) :
    print ("No Solution"
else :
    print ("Largest d such that a + b +"
        "c = d is" ,ans)
  
# This code is contributed by Manish Shaw
# (manishshaw1)

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C#

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// C# Program to find the largest
// such that d = a + b + c
using System;
  
class GFG 
      
// function to find largest d 
static int findLargestd(int []S, 
                        int n)
{
    bool found = false;
  
    // sort the array 
    // in ascending order
    Array.Sort(S);
  
    // iterating from backwards to
    // find the required largest d
    for (int i = n - 1; i >= 0; i--) 
    {
        for (int j = 0; j < n; j++)
        {
  
            // since all four a, b, c, 
            // d should be distinct
            if (i == j)
                continue;
  
            for (int k = j + 1; k < n; k++) 
            {
                if (i == k)
                    continue;
  
                for (int l = k + 1; l < n; l++) 
                {
                    if (i == l)
                        continue;
  
                    // if the current combination 
                    // of j, k, l in the set is 
                    // equal to S[i] return this 
                    // value as this would be the 
                    // largest dsince we are  
                    // iterating in descending order 
                    if (S[i] == S[j] + S[k] + S[l])
                    {
                        found = true;
                        return S[i];
                    }
                }
            }
        }
    }
    if (found == false)
        return int.MaxValue;
  
    return -1;
}
  
// Driver Code
public static void Main() 
    // Set of distinct Integers
    int []S = new int[]{ 2, 3, 5, 7, 12 };
    int n = S.Length;
  
    int ans = findLargestd(S, n);
    if (ans == int.MaxValue)
        Console.WriteLine( "No Solution");
    else
        Console.Write("Largest d such that a + " +
                          "b + c = d is " + ans );
          
}
  
// This code is contributed by Sam007

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PHP

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<?php
// PHP Program to find the largest 
// d such that d = a + b + c
  
function findLargestd( $S, $n)
{
$found = false;
  
    // sort the array in
    // ascending order
    sort($S);
  
    // iterating from backwards to 
    // find the required largest d
    for ( $i = $n - 1; $i >= 0; $i--) 
    {
        for ( $j = 0; $j < $n; $j++) 
        {
  
            // since all four a, b, c,  
            // d should be distinct
            if ($i == $j)
                continue;
  
            for ( $k = $j + 1; $k < $n; $k++) 
            {
                if ($i == $k)
                    continue;
  
                for ( $l = $k + 1; $l < $n; $l++) 
                {
                    if ($i == $l)
                        continue;
  
                    // if the current combination 
                    // of j, k, l in the set is 
                    // equal to S[i] return this 
                    // value as this would be the 
                    // largest d since we are  
                    // iterating in descending order
                    if ($S[$i] == $S[$j] + $S[$k] + $S[$l]) 
                    {
                        $found = true;
                        return $S[$i];
                    }
                }
            }
        }
    }
    if ($found == false)
        return PHP_INT_MIN;
}
  
// Driver Code
  
// Set of distinct Integers
$S = array( 2, 3, 5, 7, 12 );
$n = count($S);
  
$ans = findLargestd($S, $n);
if ($ans == PHP_INT_MIN)
    echo "No Solution" ;
else
    echo "Largest d such that a + b + "
         "c = d is " , $ans ;
  
// This code is contributed by anuj_67.
?>

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Output :

Largest d such that a + b + c = d is 12

This brute force solution has a time complexity of O((size of Set)4).

Method 2(Efficient Approach – Using Hashing)
The above problem statement (a + b + c = d) can be restated as finding a, b, c, d such that a + b = d – c. So this problem can be efficiently solved using hashing.

  1. Store sums of all pairs (a + b) in a hash table
  2. Traverse through all pairs (c, d) again and search for (d – c) in the hash table.
  3. If a pair is found with the required sum, then make sure that all elements are distinct array elements and an element is not considered more than once.

Below is the implementation in C++.

C++

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// A hashing based CPP program to find largest d
// such that a + b + c = d.
#include <bits/stdc++.h>
using namespace std;
  
// The function finds four elements with given sum X
int findFourElements(int arr[], int n)
{
    // Store sums  (a+b) of all pairs (a,b) in a 
    // hash table
    unordered_map<int, pair<int, int> > mp;
    for (int i = 0; i < n - 1; i++)
        for (int j = i + 1; j < n; j++)
            mp[arr[i] + arr[j]] = { i, j };
  
    // Traverse through all pairs and find (d -c)
    // is present in hash table
    int d = INT_MIN;
    for (int i = 0; i < n - 1; i++) {
        for (int j = i + 1; j < n; j++) {
            int abs_diff = abs(arr[i] - arr[j]);
  
            // If d - c is present in hash table,
            if (mp.find(abs_diff) != mp.end()) {
  
                // Making sure that all elements are
                // distinct array elements and an element
                // is not considered more than once.
                pair<int, int> p = mp[abs_diff];
                if (p.first != i && p.first != j && 
                    p.second != i && p.second != j)
                    d = max(d, max(arr[i], arr[j]));
            }
        }
    }
    return d;
}
  
// Driver program to test above function
int main()
{
    int arr[] = { 2, 3, 5, 7, 12 };
    int n = sizeof(arr) / sizeof(arr[0]);
    int res = findFourElements(arr, n);
    if (res == INT_MIN)
        cout << "No Solution.";
    else
        cout << res;
    return 0;
}

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Output:

12

The overall time complexity for this efficient approach is O(N2) (where N is the size of the set).



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